Lagrangian mechanics for a varible mass

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Discussion Overview

The discussion centers on the formulation of Lagrangian mechanics for systems with variable mass, exploring theoretical approaches and specific examples such as a chain falling off a table and a rocket. Participants are considering how to derive valid Lagrangians in these contexts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the possibility of finding a valid Lagrangian for a variable mass system, proposing a specific form for a chain falling over a table but seeking feedback on its validity.
  • Another participant suggests that the derivation of Lagrange's equations relies on constant mass, implying that a re-derivation for variable mass may be necessary.
  • A third participant speculates on starting the re-derivation process from D'Alembert's principle as a potential approach.
  • A later reply references external sources that discuss Lagrangians in the context of falling chains, indicating that there may be existing work on the topic.

Areas of Agreement / Disagreement

Participants express differing views on the feasibility of applying Lagrangian mechanics to variable mass systems, with no consensus reached on the correct approach or formulation.

Contextual Notes

Participants note the challenge of deriving Lagrangians for variable mass systems, highlighting the dependence on foundational principles and the need for further exploration of existing literature.

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Is it possible to find a lagrangian for a system with a varible mass and have a vaild solution when you are complete?

For instance, if I have a chain falling over the edge of a table or a rocket how would one approach this.Example of my thinking:
For a chain falling over a table, if we assume that a sufficent portion of the entire mass is hanging over the edge (M'), than the lagrange will be something along the lines of:

L=1/2 (dm/dq)*v(t)^2 + M'gl; where q is a generalized coordinate, and l is how high the object is above the ground-level.

Anyone see anything wrong with this picture? Any other examples?
 
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I would think, since the derivation of Lagrange's equation is predicated on constant mass, that you would have to go back to the beginning and re-derive the results for variable mass. In fact, that might be an interesting exercise until itself...
 
The question then becomes, I guess where to start with that.

I guess we could attempt to derive it from D'Alembert's principle, but that is just a guess.
 
I looked up some references. This one uses a Lagrangian in a slightly different context

httP://personnel.physics.ucla.edu/directory/faculty/fac_files/wong_cw/ajp_2006_falling_chains.pdf[/URL]

And Dr Math seems to have solved it here -
http://mathforum.org/library/drmath/view/56302.html
 
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